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Magazine Chapter 6: Problem 72
Solve each formula for the indicated variable. $$A=\frac{P i}{1-(1+i)^{-n}}, \text { for } n$$
Short Answer
Expert verified
\( n = \frac{-\ln\left(1 - \frac{Pi}{A}\right)}{\ln(1+i)} \).
Step by step solution
01
Rearrange the Formula
Our given formula is:\[ A = \frac{Pi}{1-(1+i)^{-n}} \]We want to solve it for \( n \). To start, rearrange the formula to isolate the fraction on one side.
02
Express in Terms of the Denominator
Multiply both sides by the denominator to eliminate the fraction:\[ A \cdot \left(1-(1+i)^{-n}\right) = Pi \]
03
Solve for Denominator Expression
Divide both sides by \( A \) to isolate the expression involving \( n \) on one side:\[ 1-(1+i)^{-n} = \frac{Pi}{A} \]
04
Isolate Power Term
Subtract the result from 1 to isolate the negative power term:\[ (1+i)^{-n} = 1 - \frac{Pi}{A} \]
05
Use Logarithms to Solve for n
Take the natural logarithm of both sides to solve for \( n \):\[ -n \ln(1+i) = \ln\left(1 - \frac{Pi}{A}\right) \]
06
Final Expression for n
Solve for \( n \) by dividing by \(-\ln(1+i)\):\[ n = \frac{-\ln\left(1 - \frac{Pi}{A}\right)}{\ln(1+i)} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging equations to isolate a variable or simplify an expression. It involves using arithmetic and algebraic operations like addition, subtraction, multiplication, division, and exponentiation.
In our exercise, we start with the equation:
In our exercise, we start with the equation:
- First, the goal is to rearrange this equation to find the variable of interest, which in this case is \( n \), present in the denominator of a fraction. To do this, you might need to apply several algebraic operations, always working step by step to maintain equality.
- Multiply both sides by the denominator to eliminate the fraction, which is a crucial step to simplify the expression and make it workable.
- Then, isolate the terms involving \( n \) by performing division and subtraction where necessary.
Logarithms
Logarithms are an essential concept in algebra and are used to solve equations where the variable is an exponent. A logarithm answers the question: "To what power must the base be raised, to yield a certain number?"
In the solution provided:
In the solution provided:
- The natural logarithm \( \ln \) is used to bring down the variable \( n \) from the exponent. The equation contained the term \( (1+i)^{-n} \), which can be simplified using logarithms.
- By taking the logarithm of both sides of the equation, the exponent \( -n \) is brought down, turning the equation into a linear form which can be more easily solved.
- Once the variable is isolated, further algebraic steps can proceed easily to derive a solution for \( n \).
Mathematical Formulas
A mathematical formula is essentially a concise way of expressing information symbolically. These formulas help describe relationships and outputs from certain inputs.
For the formula provided:
For the formula provided:
- We are concerned with solving for \( n \) within the formula \( A = \frac{Pi}{1-(1+i)^{-n}} \). This formula is used in fields such as finance for calculating payments, annuities, and interest.
- Solving this specifically for the variable \( n \) shows us how the period of time relates to other financial parameters like principal, interest rate, and total amount.
- Formulas are invaluable in structuring these relationships and calculations, enabling quick computation without the need to derive each relationship from scratch.
